Remark 3 Given \(\alpha = [0;\overline{a_1,\ldots , a_{\ell }}]\) , consider the irrational \(\beta =[0;\overline{a_2,\ldots ,a_{\ell },a_1}]\) , which is the image of \(\alpha \)  via the Gauss map [1]}. Relations (REF ) and (REF ) imply that
\(G_1(\beta ,\varepsilon )= G_2(\alpha ,\varepsilon ), \quad G_2(\beta , \varepsilon )= G_3(\alpha ,\varepsilon ) ,\quad \ldots \, ,\, \quad G_0(\beta ,\varepsilon ) = G_1(\alpha , \varepsilon ).\) 
